By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper, we extend the scope of this isomorphism in several directions. First, we give a direct definition of Rabinowitz loop homology in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to positive Floer homology, resp. loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce reduced loop homology, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo–Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism using linear Hamiltonians and the square root of the energy functional.

根据著名的 Viterbo 定理，闭流形余切丛的辛同调与其环空间的同调同构。在本文中，我们在几个方向上扩展了这种同构的范围。首先，我们根据环空间上的Morse理论给出了Rabinowitz环同调的直接定义，并证明其乘积与Rabinowitz Floer同调上的裤子积一致。该证明使用了穿孔环的紧化模空间。其次，我们证明，当限于正Floer 同源性时，resp。循环空间同源性相对于常量循环，Viterbo 同构将次级裤子副积的各种结构与循环同源余积交织在一起。三、我们介绍约化环同调，它是环积的规范约简和环同调余积的扩展的共同定义域，它们共同定义了交换互换单位无穷小反对称双代数的结构。在此过程中，我们证明了从二次哈密顿量的 Floer 复数到能量泛函的莫尔斯复数的 Abbondandolo–Schwarz 准同构可以使用线性哈密顿量和能量泛函的平方根转化为过滤链同构。